Turbine, positive displacement and Coriolis mass flow meters

Why it matters

When money changes hands — fuel at a depot, crude at a terminal, a batch of ingredient into a reactor — flow must be totalised accurately. Turbine and positive displacement (PD) meters count volume with typical accuracies of a few tenths of a per cent, and Coriolis meters measure mass flow and density directly, which is what chemical recipes and custody transfer actually need. Knowing what each one truly measures, and what upsets it, is basic to specifying a meter.

Key ideas

Turbine flow meter.

• A free-running bladed rotor on bearings, aligned with the pipe axis, spins at a speed proportional to the mean velocity. A magnetic or inductive pick-up outside the pipe produces one pulse per blade pass.
• Output: a pulse frequency f proportional to volumetric flow, f = K·Q, where K (the K-factor, pulses per m³ or per litre) comes from calibration. Totalising the pulses gives volume directly.
• Linear range about 10:1 or more for clean, low-viscosity liquids and gases; accuracy ±0.25 % to ±0.5 % of reading in its linear range.
• K changes at low flow (bearing friction and fluid drag) and with viscosity, so turbines are calibrated on the fluid or at the viscosity of use. Needs clean fluid (strainers), straight runs or a flow straightener to remove swirl, and protection against overspeed (gas slugs in liquid lines destroy bearings).

Positive displacement meters.

• Mechanically trap and pass discrete volumes: oval-gear, rotary-piston, nutating-disc, lobed-impeller, sliding-vane and diaphragm (domestic gas) meters.
• Flow is the displaced volume per cycle times the cycle rate: Q = Vc·n. The reading is almost independent of velocity profile, so no straight runs are needed.
• Accuracy is best with viscous liquids, because leakage (slip) through the clearances falls as viscosity rises; low-viscosity fluids slip more at low flow.
• Need clean fluid (solids jam or wear the clearances), cause a moderate pressure drop, and can block the line if they seize (bypass arrangements are used).

Coriolis mass flow meter.

• One or two tubes (U-shaped or straight) are vibrated at their natural frequency by a drive coil. Fluid flowing through a tube that is rotating or oscillating experiences the Coriolis acceleration 2·ω × v; the reaction forces on the inlet and outlet halves are opposite, so the tube twists in proportion to the mass flow.
• Pick-off coils at the inlet and outlet measure the tube motion; the twist appears as a time (phase) lag Δt between them. Δt is proportional to mass flow: ṁ = FCF·Δt (flow calibration factor from the factory).
• The resonant frequency depends on the mass of the tube plus its contents, so the same meter measures fluid density. A temperature sensor corrects the tube stiffness. Volumetric flow is then ṁ/ρ.
• Direct mass flow independent of fluid properties and velocity profile; no straight runs; accuracy around ±0.1 % of reading for liquids; handles viscous fluids and many slurries.
• Limitations: cost, size and weight in large line sizes, pressure drop, sensitivity to external vibration and to entrained gas (two-phase flow), and erosion with abrasive slurries.

Choice. Clean light hydrocarbons and gases: turbine. Viscous oils, fuel dispensing: PD. Mass-based batching, density measurement, custody transfer of liquids: Coriolis.

Formulas

f = K·Q (turbine; K in pulses/m³) and V_total = N/K (N = total pulses)

Q = Vc·n (PD meter)

F_c = 2·Δm·ω·v (Coriolis force on a fluid element of mass Δm moving at v in a frame rotating at ω, v ⟂ ω)

M = 4·ω·ṁ·L·r (twisting moment on a U-tube with legs of length L spaced 2r apart)

Δt ≈ 8·ṁ·r² / Ks (time lag between pick-offs; Ks = torsional stiffness of the tube)

ṁ = FCF·Δt (in practice, with the factory flow calibration factor)

ρ = (m_t/V)·[(f₀/f)² − 1] (density from resonant frequency; f₀ = frequency with the tube empty, m_t = vibrating tube mass, V = internal volume)

Q = ṁ/ρ

Symbols: f = pulse or resonant frequency (Hz); K = K-factor (pulses/m³); Q = volumetric flow (m³/s); Vc = volume per cycle (m³); n = cycles per second (s⁻¹); ω = angular velocity of the tube motion (rad/s); ṁ = mass flow (kg/s); L, r = tube dimensions (m); Ks = torsional stiffness (N·m/rad); FCF = flow calibration factor (kg/s per s of lag); ρ = density (kg/m³). The density formula follows from f ∝ 1/√(m_t + ρV).

Worked examples

Example 1 (standard): turbine meter. Given: K-factor = 25 pulses/L; pick-up frequency f = 200 Hz.

1. Q = f/K = 200/25 = 8.0 L/s.
2. Convert: 8.0 L/s × 3.6 = 28.8 m³/h.
3. Pulses in one hour = 200 × 3600 = 720 000, i.e. 720 000/25 = 28 800 L.
4. Answer: Q = 8.0 L/s = 28.8 m³/h.

Example 2 (GATE level): Coriolis meter giving mass flow, density and volume flow. Given: with the tube empty (air, ρ ≈ 0) the tube resonates at f₀ = 100.0 Hz; filled with water (1000 kg/m³) it resonates at 81.65 Hz. With the process liquid it resonates at f = 84.0 Hz and the pick-off lag is Δt = 2.4 µs. Flow calibration factor FCF = 1.5 kg/s per µs.

1. From the water point: (f₀/f_w)² − 1 = (100/81.65)² − 1 = 1.500 − 1 = 0.500, so m_t/V = 1000/0.500 = 2000 kg/m³.
2. Process liquid: (f₀/f)² − 1 = (100/84.0)² − 1 = 1.4172 − 1 = 0.4172.
3. ρ = 2000 × 0.4172 = 834.5 kg/m³.
4. ṁ = 1.5 × 2.4 = 3.60 kg/s.
5. Q = ṁ/ρ = 3.60/834.5 = 4.314 × 10⁻³ m³/s = 15.5 m³/h.
6. Answer: ρ ≈ 834 kg/m³, ṁ = 3.6 kg/s, Q ≈ 15.5 m³/h.

Example 3 (PD meter). An oval-gear meter displaces 50 cm³ per revolution and turns at 300 rpm. Q = 50 cm³ × 300 /min = 15 000 cm³/min = 15 L/min.

Common mistakes

• Treating the turbine K-factor as constant at very low flow or for a fluid of different viscosity.
• Multiplying instead of dividing: Q = f/K when K is in pulses per unit volume.
• Saying a Coriolis meter measures a "frequency shift" proportional to flow: the flow signal is the phase or time lag; the frequency gives density.
• Using a Coriolis meter on a line with entrained gas or strong pipe vibration without checking the specification.
• Choosing a PD meter for a dirty fluid, or a turbine for a viscous oil.
• Forgetting that a turbine or PD meter measures volume at line conditions: converting to mass needs density at those conditions.

For GATE IN

Questions are mostly conceptual: which meter measures mass directly, why PD meters suit viscous fluids, why turbines need clean fluids and straight runs, and what the Coriolis meter's phase lag and resonant frequency represent. Numericals: flow from pulse frequency and K-factor, PD volume per cycle, Coriolis force or mass flow from a given factor, and mass-volume conversion with density.

Quick check

1. A turbine meter with K = 400 pulses/m³ produces 120 Hz. What is the flow in m³/h?
2. Which Coriolis measurement gives density?
3. Why do PD meters become more accurate with more viscous liquids?
4. A PD meter of 0.5 L per cycle runs at 4 cycles/s. What is the flow?
5. Name two things that degrade Coriolis-meter accuracy.

Answers: 1. 120/400 = 0.3 m³/s = 1080 m³/h. 2. The tube's resonant frequency. 3. Slip through the clearances falls as viscosity rises. 4. 2 L/s. 5. Entrained gas and external vibration (also erosion or coating of the tube).